Transfer Prices for Multienterprise Supply Chain Optimization

Mar 3, 2001 - A key issue in supply chain optimization involving multiple enterprises is the determination of policies that optimize the performance o...
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Ind. Eng. Chem. Res. 2001, 40, 1650-1660

Transfer Prices for Multienterprise Supply Chain Optimization Jonatan Gjerdrum,† Nilay Shah,*,† and Lazaros G. Papageorgiou†,‡ Centre for Process Systems Engineering, Imperial College, London SW7 2BY, U.K., and Department of Chemical Engineering, University College London, London WC1E 7JE, U.K.

A key issue in supply chain optimization involving multiple enterprises is the determination of policies that optimize the performance of the supply chain as a whole while ensuring adequate rewards for each participant. In this work, a mathematical programming formulation is presented for fair, optimized profit distribution between members of multienterprise supply chains. The proposed formulation is based on a novel approach applying game theoretical Nash-type models to find the optimal profit level for each enterprise subject to given minimum profit requirements. A modeling framework for distributed profit optimization for an n-enterprise supply chain network is first presented. The supply chain planning problem is then formulated as a mixedinteger nonlinear programming model including a nonlinear Nash-type objective function. Model decision variables include intercompany transfer prices, production and inventory levels, resource utilization, and flows of products between echelons, subject to a deterministic sales profile, minimum profit requirements for each enterprise. and various resource constraints. A separable programming approach is finally applied utilizing logarithmic differentiation and approximations of the variables of the objective function. The resulting model is of the mixed-integer linear programming form. The applicability of the approach is demonstrated through case studies based on industrial processes relevant to process systems engineering. 1. Introduction Conflicting interests in general extended multienterprise supply chains frequently lead to problems in how to distribute the overall value to each member of the supply chain. A simple approach to enhance the performance of a multienterprise supply chain is to maximize the summed enterprise profits of the entire supply chain subject to various network constraints. When the overall system is optimized in this fashion, there is no automatic mechanism to allow profits to be fairly apportioned among participants. Solutions to this class of problems usually exhibit quite uneven profit distributions and are therefore impractical. They do, however, give an indication of the best possible total profit attainable in the supply chain as well as an indication of the best activities to carry out. The aspiration of this work is to extend the above approach so as to achieve solutions of similar quality overall but with equitable profit distributions. Power relations exist in the supply chain, and every member seeks to leverage as much value to themselves as possible. For example, manufacturing companies in a dominant role relative to other supply chain members may gain a structural advantage, rendering opportunities for attaining higher profits than their suppliers and/ or distributors. Possession of power is demonstrated by a relative capacity by owners of particular resources to safeguard value when participating in the supply chain. The distribution of physical products in the supply chain exists in an exchange relationship with the value chain, which distributes the revenue stream from customers to each member of the supply chain. The problem of * Author to whom all correspondence should be addressed. E-mail: [email protected]. Phone: +44 20 7594 6621. Fax: +44 20 7594 6606. † Imperial College. ‡ University College London.

distributing profits to each member of the supply chain is illustrated in Figure 1 (adapted from Cox1). The aim is to find solutions to this class of problems given a quantitative prior power structure of the supply chain. It should be clear that the companies involved in the supply chain are in business only to create value for themselves. For example, sharing of information between partners will only take place if the supply chain players believe their profits will be leveraged. The underlying assumption is that the surrounding business environment is suitable for long-term partnerships and that the power relations between the companies can somehow be quantified in terms of minimum acceptable profit levels. These levels are clearly dependent on the customer market for the products manufactured in the supply chain as well as the vertical intercompany power relations in the supply chain. It should be added that these power relations may be affected by the existence of external competitors. van Hoek2 raises the question of how to divide supply chain revenues among the players in a supply chain system when there is no leading player to determine how distribution of benefits should be handled. He states that supply chain control is no longer based on direct ownership but rather on integration over interfaces of functions and enterprises. Traditional performance indicators limit the possibilities of optimizing the supply chain network because the measures do not correctly address the wide opportunities for improvement. The supply chain measurement system should be able to give each member an indicative measure before entering of, for example, how much return on investment can be expected from being a part of a particular supply chain. In light of these problems, this paper aims at obtaining quantitative values of a fair profit objective for each company in the supply chain based on the available supply chain resources. Though the effectiveness of operations has improved lately because of the introduction of enterprise resource

10.1021/ie000668m CCC: $20.00 © 2001 American Chemical Society Published on Web 03/03/2001

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Figure 1. Relation between the supply chain and the value chain.

planning systems and so-called advanced planning systems, there is still wide interest in rigorous optimization of supply chains delivering solutions of large-scale problems. The optimization approach presents the opportunity of guaranteeing optimal or near-optimal solutions within a specified margin. Substantial results have been obtained in the literature over the last 10 years in this area as illustrated in the rest of this section. Cohen and Lee3 deliver a model for making resource deployment decisions in a global supply chain network and solve different scenarios using an extensive mixedinteger nonlinear programming (MINLP) model. The use of “structural solutions”, i.e., discrete alternative solutions, reduces the model to linear form, a mixedinteger linear programming (MILP) model. They also discuss several “policy options” for plant utilization, supply, and distribution strategies. The objective function and network constraints are quite detailed and contain terms such as taxes paid in different countries and requirement levels to establish manufacturing facilities in a country. After developing a very large mixed-integer linear global supply chain model, Arntzen et al.4 conclude that “we also wonder how anyone can rely on heuristic solution methods in this arena”. The model minimizes cost or distribution times, meeting estimated demand, local restrictions, offset trade, and maximum capacity. The model was used to restructure the Digital Equipment Corp., and the authors claim it saved the company over $100 million. Recently, a few papers have appeared that describe ways of handling supply chain transfer prices. Pfeiffer5 describes transfer pricing in a supply chain consisting of procurement, manufacturing, and selling units of one single company. His theoretical model handles one commodity at each node and does not include any capacity constraints. He proposes a transfer price system governed by the headquarters, which fixes a specific transfer price level. Each node optimizes its own decisions independently to maximize a given profit function, according to the price level fixed by the headquarters. After the decentralized optimization, headquarters evaluates and collects the overall results obtained and chooses a new transfer price which leads to a higher overall profitability. A Lagrangean dualdecomposition approach is utilized with each problem as a linear program. This algorithm stops when the

overall profit function is maximized. In common with the model proposed in this paper, Pfeiffer uses the transfer price as a value measure of the products stored. As would be expected, his conclusion is that the stock level and the production intensity (output/time) in the process system should be chosen to be minimal, thus minimizing inventory and production costs subject to production requirements. Alles and Datar6 claim that companies choose their cost-based transfer prices based on their competitive environment. The enterprise may cross-subsidize products in order to increase their ability to increase prices. Often, transfer prices for relatively lower cost products are increased, while those of higher cost products are decreased. The authors give evidence that transfer prices are determined based on strategic decisions rather than on internal cost systems. Jose and Ungar7 propose an approach to decentralized pricing optimization of interprocess streams in chemical industry companies. Their iterative auction method determines the prices of process streams so as to maximize an objective for a single chemical company, while each division within the company is constrained by its available resources. The approach is interesting in that each division conceals its private information from the other parties within the so-called “micro supply chain”. It normally takes several iterations for a model to converge, and the user has to define the limited amount of slack resources utilized. One of the main conceptual differences between their approach and the one presented in this paper is that they regard the channel members as adversarial and competitive for resources rather than cooperative. Also, they use a slack resource iterative auction approach, whereas in this paper the solution approach is to solve a noniterative separable MILP problem. Ballou et al.8 stress the importance of common objectives in the entire supply chain. Unattainable improvements for single companies in terms of cost savings and customer service enhancements can be obtained by cooperative companies. The authors point out that problems arise if some of the firms benefit at the expense of the others. The conflict resolution between supply chain partners must thus be of focal interest, and to keep the coalitions intact, the rewards of cooperation must be redistributed. They identify three means to achieve this: (1) Metrics could be developed

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to capture the nature of interorganizational cooperation to simplify benefit analysis. (2) Information sharing mechanisms could transfer information about the benefits of cooperation among the members in the supply chain. (3) Allocation methods could be developed that fairly distribute the rewards of cooperation between the members. We attempt to address these three issues quantitatively in the present paper. An approach for simultaneously optimizing the profits in multienterprise supply chain partnerships is proposed in which the separate firms try to optimize their profits fairly given their lower profit requirement levels as well as their technology standard. We first present a MINLP approach to determine the most appropriate transfer price level for products transferred between production sites as well as from production sites to distribution centers. The MINLP model is tackled in a separable programming approach in which the utilization of logarithmic differentiation and approximations of the variables of the objective function result in a model of the MILP form. The solution obtained includes the transfer prices, production levels, timing and quantities of products delivered, and resources utilized. The inventory cost at the production plant depends on the transfer price level and is therefore not arbitrary in regards to the overall supply chain optimization. The paper is organized as follows. In section 2, the general problem is briefly described with relevant assumptions. Given parameters and required data as well as the model objectives and the variables to be determined are listed. In section 3, the mathematical model is presented. The model constraints are divided into production resource, inventory, demand, cost, transfer price, and linearization constraints categories. Then, in section 4, the Nash approach is briefly described. Why this particular approach is favored for the problem under consideration is explained. In section 5, several computational experiments are described and evaluated. Some concluding remarks are drawn in section 6. Finally, in the appendix, the utilized separable approach is described. 2. Problem Description A general supply chain is considered, which is composed of N different enterprises. There are three different types of nodes in the supply chain. A “primary” company produces intermediate products which are delivered to a “secondary” company, where the intermediates are converted to final products. Finally, a “tertiary” company is a warehouse or distribution center (where no production takes place) from which the final products are delivered to customers. In any company of the supply chain, product i can be produced utilizing R production resources. In secondary companies, products are produced using production resources and intermediate products purchased from primary companies, and then the products are supplied downstream to tertiary companies. The prices for products shipped between primary and secondary companies and secondary and tertiary companies are called transfer prices. The decisionmakers in the primary, secondary, and tertiary companies have K discrete transfer price levels to choose from. Inventory can be held in all companies. Costs include raw material, manufacturing, administrative, inventory, purchase, and transportation cost. The revenue is determined by sales prices or transfer prices multiplied by the amount shipped to

Figure 2. Time horizon discretization.

downstream nodes. The revenues less the corresponding costs constitute the objective function (“profit”) for each company. The following assumptions have been made: (a) All information is available for the entire overall supply chain optimization. (b) External demand and external sales prices are deterministic over a short to medium term time horizon. (c) Alternative transfer price levels are available to choose from. (d) A long-term supply chain partnership is established, indicating that decisionmakers can accept transfer prices which may or may not be the current market prices of the transferred products. The overall problem can formally be stated as follows: Given (a) time horizon, (b) external demand, (c) external sales prices, (d) raw material costs, (e) production data, (f) inventory data, (g) transportation data, (h) minimum acceptable profit requirements, and (i) range of available transfer prices, determine (a) production plan, (b) inventory policies, (c) transportation plan, and (d) transfer price levels, so as to (a) find multienterprise strategies which result in optimal, fair profitability of the enterprises and (b) fulfill demand. 3. Mathematical Formulation To facilitate the mathematical model, a time discretization approach is applied over a planning horizon, H, of T discrete time intervals of equal duration, δ, as illustrated in Figure 2. The key variables in the formulation include production levels in primary companies and secondary companies, inventory levels in all companies, flow levels of products between primary and secondary companies and secondary and tertiary companies, and transfer price levels for the product flows. These are determined so as to maximize the profits of all companies, subject to production resource constraints, transportation constraints, inventory constraints, transfer price level constraints, mass balances, and cost constraints. The model is described in detail below. First, the following notation is introduced. Indices m ) member i, j ) products r ) resource k ) transfer price available t ) time Sets Rm ) resources of m Pr ) set of products using r SMi ) set of members selling i T h im ) set of members receiving i from m Tim ) set of members providing i to m Om ) set of products produced by m Parameters (m denotes member in all definitions below) Rimr ) demand for resource r for production of product i Amrt ) capacity for resource r at time t Fijmr ) unit consumption of resource r and intermediate product i for the production of product j Dimt ) customer demand quantity of product i at time t

Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1653 r ) inventory holding cost constant δ ) time period duration (in days) ηim ) market sales price of product i HC0im ) handling cost constant of product i 0 MCimr ) manufacturing cost constant, using resource r for production of i PC0im ) purchase cost constant of product i TC0im ) transportation cost constant of product i P ˆ imm′k ) k available transfer price levels for product i Binary Variables

Continuous Variables Bimt ) batch size of product i at time t Iimt ) inventory level of product i at time t Fimm′t ) amount shipped of product i between members m and m′ at time t Simt ) amount shipped of product i to customers at time t ICm ) inventory cost Pimm′ ) transfer price of product i between members m and m′ HCm ) handling cost MCm ) manufacturing cost PCm ) purchase cost TCm ) transportation cost Cm ) overall cost πm ) profit

3.1. Production Resource Constraints. The amount of each production resource utilized over any period t at the production site cannot exceed the plant availability level, Amrt, in that period. Thus, for each member m the utilization of production resource r for each product i is RimrBimt, where Rimr is the unit consumption of resource r for production of one unit of product i and Bimt is the amount produced of product i at time t. The utilization of production resources for each member m, resource r, and time interval t should be less than the resource availability level, Amrt.

∑ RimrBimt e Amrt

∀ m, r ∈ Rm, t

(1)

i∈Pr

3.2. Inventory Constraints. The inventory level of product i at time t for each member m, Iimt, is equal to the amount at time t - 1, Iim,t-1, added to the amount produced by member m, Bimt, added to the amount received, ∑m′∈TimFim′mt, less the amount used to produce other products by consuming product i, ∑j∈Om∑r∈RmFijmrBjmt (Fijmr is the unit consumption of resource r and intermediate i to produce a downstream product j), and the amount shipped to other members, ∑m′∈Th imFimm′t, and the amount shipped to customers, Simt. Hence,

Iimt ) Iim,t-1 + Bimt +

∑

Fim′mt -

∑ ∑

j∈Om r∈Rm

rδ

∑i ∑t 365 Iimt(m′∈T ∑h

∀ m (4)

Pimm′ + ηim)

FijmrBjmt -

∑

m′∈T h im

3.4.2. Administrative Cost. The cost of handling product i for member m at time t is a constant, HC0im, multiplied by the total amounts handled, be it amount produced, Bimt, amount sold, Simt, incoming amount, ∑m′∈TimFim′mt, or outgoing amount, ∑m′∈Th imFimm′t. For each member, two of the terms will normally be nonzero, i.e., one incoming amount and one outgoing amount.

HCm )

∑i ∑t (Bimt + Simt + m′∈T ∑h

Fimm′t +

∑

Fim′mt)

m′∈Tim

im

∀ m (5)

3.4.3. Manufacturing Cost. The manufacturing cost for producing i is a constant, MC0imr (depending on the resources utilized), multiplied by the amount manufactured, Bimt:

MCm )

∑i ∑t r∈R ∑ MC0imrBimt

(6)

m

3.4.4. Purchase Cost. The purchase cost for member m consists of the costs of raw materials purchased from external producers (given by a constant cost for each raw material i, PC0im, multiplied by the usage, Bimt). Furthermore, intermediate products can also be purchased from other members m′. In this case, the purchase cost is obviously the price of the intermediate product, Pim′m, multiplied by the amount purchased, Fim′mt, summed over all members providing the intermediate product.

PCm )

∑i ∑t (PC0imBimt + m′∈T ∑

Pim′mFim′mt)

im

∀m (7)

3.4.5. Transportation Cost. The transportation cost is determined by the amount shipped from member m either to customers, Simt, or to other members, ∑m′∈Th imFimm′t. The sum is multiplied by a product transportation constant, TC0im.

TCm )

∑i ∑t TC0im(Simt + m′∈T ∑h

Fimm′t)

∀ m (8)

im

Fimm′t - Simt ∀ i, m, t (2)

3.3. Demand Constraints. An external demand constraint is used for all products i which is to fulfill variable demands at all time intervals t. Simt is the shipped quantity of product i at time t. Dimt is the corresponding customer demand of product i at time t.

Simt e Dimt

ICm )

im

Ximm′k ) 1, if the kth price level is chosen between members m and m′ for product i; 0, otherwise

m′∈Tim

3.4. Definition of Costs. 3.4.1. Inventory Cost. The inventory cost depends on the outgoing prices associated with the member m: ∑i∑t∑m′∈Th im(rδ/365)IimtPimm′ if other members m′ consume products i or ∑i∑t(rδ/365)Iimtηim (where ηim is the constant selling price to customers over the time horizon) if products are sold to customers. Consequently,

∀ i, m ∈ SMi, t

(3)

3.4.6. Overall Cost Constraints. The overall cost is simply the sum of the inventory cost, the administrative cost, the manufacturing cost, the purchase cost, and the transportation cost. This amount constitutes the total cost base for member m.

Cm ) ICm + HCm + MCm + PCm + TCm

∀ m (9)

3.5. Profit Definition Constraints. The profit definition constraints simply state that the profit of each member of the supply chain is the revenue less the cost

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incurred at that supply chain node. The revenue is either the selling price to customers, ηim, multiplied by the amount sold, Simt, or the transfer price to other consuming members m′, Pimm′, multiplied by the amount shipped, Fimm′t. The profit, πm, of each member, m, is then defined by

πm )

∑i ∑t (ηimSimt + m′∈T ∑h

Pimm′Fimm′t) - Cm

im

∀m (10)

3.6. Transfer Price Levels. It is assumed that K discrete transfer price levels for each product i transferred between members are defined via the parameter P ˆ imm′k. To determine the price level Pimm′, the binary ˆ imm′k and decision variable Ximm′k is multiplied by P summed over all transfer price levels for that product. For each product i, only one transfer price level is chosen because ∑kXimm′k ) 1.

Pimm′ )

∑ Pˆ imm′kXimm′k k∈TP

∀ i, m, m′ ∈ T h im (11)

i

∑k Ximm′k ) 1

∀ i, m, m′ ∈ T h im

(12)

3.7. Equivalent Linear Formulation. There are nonlinearities in constraints (4), (7), and (10), respectively. This is due to the product of the transfer price, Pimm′, and inventory level, Iimt, and the product of the transfer price and the amount shipped, Fimm′t. Therefore, we introduce the following variables to convert the nonlinear model above to an exact linear equivalent: XIimm′kt t Ximm′kIimt and XFimm′kt t Ximm′kFimm′t. An appropriate upper bound for the amount shipped of each product i from member m to m′ ∈ T h im is U , and for the inventory level of product introduced, Fimm′ U i for each member m, IU im, where Iim ) upper bound on U inventory for product i and Fimm′ ) upper bound on flow of product i between members m and m′. The following linearization constraints are introduced:

∀ i, m, m′ ∈ T h im, k, t (13)

XIimm′kt e Ximm′kIU im Iimt )

∑ ∑

m′∈T h im k∈TPi

XIimm′kt

U XFimm′kt e Ximm′kFimm′

Fimm′t )

∑

XFimm′kt

∀ i, m, t

(14)

∀ i, m, m′ ∈ T h im, k ∈ TPi, t (15) ∀ m, m′ ∈ T h im, i, t (16)

k∈TPi

ˆ imm′kXFimm′kt is used inConsequently, the term ∑kP stead of the nonlinear term Fimm′tPimm′. Also, the nonˆ imm′k linear term Pimm′Iimt is replaced by ∑m′∈Th im∑kP XIimm′kt. 4. Nash Approach A very simple, naive, single-level approach of optimizing the supply chain system can be to maximize the summed profits of all players, i.e., max Φ ) ∑mπm. This approach is easy to handle computationally, but it may lead to an unfair profit distribution and indeed does so for the examples studied in this work. Hence, an

Figure 3. Nash model.

approach will be required that can produce a fair profit distribution subject to similar overall performance. The Nash solution,9 as illustrated in Figure 3 for two players, determines the fair bargaining solution positions of the different players based on their “original” position. The Nash model is based on four axioms that a rational bargaining solution should obey. These are characterized by Pareto optimality, symmetry, scale invariance, and independence of irrelevant alternatives.10 The solution delivers an optimal fair split of payoff to each of the rational players in a game, and the Nash approach is therefore a suitable method to solve the particular type of problems under consideration. The generalized Nash-Harsanyi11 extension to the Nash solution proposes that if each member has a lower profit requirement point, xLm, then the fair solution point is xm, xm g xLm, which maximizes Φ ) ∏m(xm xLm). If the minimum acceptable profit levels for, say, a three-player game are πLm (m ∈ {A, B, C}), then the Nash solution is the point πm with πA g πLΑ, πB g πLB, and πC g πLC, which maximizes the product (πA πLA)(πB - πLB)(πC - πLC). The Nash model is generally accepted as a normative value measure.12 On the basis of the Nash approach, Gjerdrum et al.13 proposed a spatial branch and bound procedure which resulted in a more sustainable split of profit between two or three participating supply chain enterprises. They applied a spatial and binary variable branch-andbound procedure to the Nash problem formulation based on approximate linearizations of the bilinear or trilinear terms involved in the model, while at each node of the search tree a MILP problem is solved. To summarize, the proposed Nash problem is inherently nonlinear and nonconvex because of products of continuous variables appearing in the objective function. The problem statement subject to constraints is to maximize the objective function Φ ) ∏m(πm - πLm), with πm defined in constraint (10) subject to production resource constraints (1), inventory constraints (2), demand constraints (3), inventory cost constraints (4), administrative cost constraints (5), manufacturing cost constraints (6), purchase cost constraints (7), and transportation cost constraints (8). The transfer prices are defined via discrete decision variables according to constraints (11) and (12). The resulting nonlinearities in constraints (4), (7), and (10) are replaced by the linearization variables defined in constraints (13), (14), (15), and (16).

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Generally, separable programming approaches express nonlinear objective functions and constraints as a sum of functions involving only one variable. For the details of how the separable approach has been implemented in the proposed model, see the appendix. 5. Computational Experiments

Figure 4. Piecewise linear approximation of a function f.

The nonlinearities in the objective function are tackled in this work via a separable programming approach to linearize the problem, i.e., the maximization of the Nash objective function in which for each company (member m) the lowest acceptable profit level is given. If the lower profit requirement point is suboptimal, then the Nash algorithm finds the solution of the problem that most fairly (subject to the Nash axioms) distributes the surplus amount obtained by the problem optimization. The solution approach applied is based on a separable technique that is described in the appendix.

In the following experiments, GAMS14 is used as the modeling environment. The MILP solver used is CPLEX15 using standard branch and bound strategy. For all computational runs, a Sun SPARC Ultra 60 is used with a margin of optimality of 5%. For all solutions, the relative error is defined as the relative difference between the actual and the approximated value: relative error ) (Φ - Φ ˆ )/Φ, where Φ ) ∏m(π/m - πLm) at the optimal solution point π/m. In Figure 4, the error depicted is the difference between the actual and the approximated value i.e., f(λ*) less ˆf(λ*). 5.1. A Two-Enterprise Example. A two-enterprise example is considered with one plant manufacturing final products (secondary company) and one distribution center (tertiary company). In the production plant, the

Figure 5. Two-enterprise example: state-task network for a production plant.

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Figure 6. Two-enterprise example: usage of a resource tower at a plant.

material in the first step of the process is a liquid feed which is dried in a tower to obtain four different powders which, in turn, are packed in various sizes to make 18 final products, of which 15 are currently in demand. They belong to four different families, based on the intermediate used for production. The plant production process is illustrated in Figure 5. The packing of the goods uses six different packaging lines and a limited amount of manpower and material handling resources. The base feed (“Feed”) is converted into four intermediate products (“Int1” to “Int4”) by four separate processes (“Proc1” to “Proc4”). The intermediate products are then sent to packaging (“Packline1” to “Packline6”). Final products [“P11” to “P43” (products P13, P23, and P33 are currently not required by customers)] are obtained by using different packaging lines in order to make products of different shape and size. The distribution center output restrictions are characterized by an irregular deterministic demand profile for a time horizon of 8 weeks with a time discretization interval of 1 day and fixed sales prices throughout this period. As is quite typical within industrial systems today, products are delivered downstream at the end of each week in the planning period, i.e. days 7, 14, 21, etc. The following quantities of the production and distribution problem are determined while maximizing the profit levels of the separate enterprises: production resource utilization, production levels, inventory levels, flows, and transfer prices of the products. The production resource utilization for the tower resource over the production horizon is displayed in Figure 6. The maximum availability each day is 1150 units processed. It can be seen that the resource is at full capacity (at the upper bound) before each customer delivery (e.g., t ) 6, 7, 13, 14) and that the tower is not utilized at all during 10 time intervals (e.g., t ) 1, 2, 8, 9, 21). To utilize this policy leads to minimized inventory cost and is therefore favored to a policy in which production is evenly spread over the horizon. The transfer price levels for five of the 15 products (products P25, P31, P32, P34, and P35) established by the optimization are displayed in Figure 7. As can be seen, some of the product prices take the minimum and maximum values in the available range while other product prices take on intermediate values. The optimal transfer price levels determined result in the enterprise profits shown in Figure 8. The first set of bars shows the lower profit

Figure 7. Two-enterprise example: transfer prices for five products.

Figure 8. Two-enterprise profits before and after supply chain optimization.

requirements of the two enterprises and the total supply chain profit, ∑π, prior to the optimization of the system. The second set of bars shows the profits obtained after the Nash optimization approach is used. The profits for the secondary and tertiary companies are 17 786 and 37 890, respectively, given lower profit requirement levels of 5000 and 25 000. The λ multipliers take nonzero values for λ31 ) 0.282, λ41 ) 0.718, λ12 ) 0.001, and λ22 ) 0.999. As can be seen, two adjacent λ’s take nonzero values as is guaranteed by the separable approach. The error at the solution point is 0.013%, and the CPU requirement is roughly half an hour; see Table 1. The naive approach would have given profit levels of 5065 and 52 480, respectively. The result of the proposed approach implies an equitable profit split and is more sustainable for the supply chain as a whole. Table 1 gives the problem size for the different examples, the marginal gaps used when solving the problems, the CPU time required on a Sun SPARC Ultra 60, and finally the relative error of the solutions obtained. The solution error is under 1% for all ex-

Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1657 Table 1. Computational Performance for Four Examples example

time period (weeks)

binaries

continuous

constraints

gap (%)

CPU (s)

relative error (%)

2 enterprises 3 enterprises 5 enterprises 6 enterprises

56 42 42 42

150 120 170 340

25 354 14 217 26 945 51 039

27 692 9 777 22 157 38 241

5 5 5 5

2115 542 19 646

0.013 0.60 0.20 0.023

Figure 9. Three-enterprise example: usage of a resource tower at A.

amples, which is to be considered a favorable result for the overall approach. 5.2. A Three-Enterprise Single Sourcing Example. In the three-enterprise example, a base chemical manufacturing plant (primary company A) is linked to a downstream specialty chemical plant (secondary company B), which in turn is linked to a distribution center (tertiary company C). A horizon of 6 weeks using 1-day time discretization is modeled in this example. The manufacturing problem in the primary company is the following. Material in the first step of the process is a feed which is dried in a tower to four different powders, which in turn are packed to make five intermediate products. In the secondary plant, the intermediate products are mixed with specific amounts of additives to produce five final, unpacked products which are then packaged. The packaging of the products in the secondary company comprises three packaging lines and a limited amount of manpower and material handling resources. The two final packed products (made up of the five unpacked products) are shipped downstream to the tertiary company. The tertiary company faces a deterministic demand and fixed sales prices for the time horizon of 6 weeks. The production resource utilization for the tower in primary company A is displayed in Figure 9. The maximum resource availability is 1000 units. The tower is mainly utilized before a customer delivery as in the previous example. Figure 10 shows the transfer prices of intermediate products (products P1 to P5) between A and B. The transfer prices span the whole available range of possible values. The transfer price levels determined result in the enterprise profits shown in Figure 11. The solution implies that all companies increase their profits by a fair amount. The profits for A-C are 4757, 9218, and 12 365, respectively, given lower profit requirement levels of 1000, 3000, and 7000. The λ multipliers take nonzero values for λ11 ) 0.792, λ21 ) 0.208, λ32 ) 0.782, λ42 )

Figure 10. Three-enterprise example: transfer prices between A and B.

Figure 11. Three-enterprise profits before and after supply chain optimization.

0.218, λ43 ) 0.954, and λ53 ) 0.046. The CPU requirement is slightly less than 10 min, and the error at the solution point (which again is the difference between the piecewise linear approximation of the objective function to maximize in the separable approach and the actual function value) is 0.60%. 5.3. A Five-Enterprise Example. The five-enterprise example is similar to the three-enterprise example except that there are three specialized producers of intermediate products, A1, A2, and A3 (primary companies). For some of the products, they compete based on different production costs, and for others, they are the only suppliers that have the required production resources available. The connectivity network is given in Figure 12.

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Figure 14. Six-enterprise connectivity. Figure 12. Five-enterprise connectivity.

Figure 15. Six-enterprise profits before and after supply chain optimization. Figure 13. Five-enterprise profits before and after supply chain optimization.

The determined transfer price levels result in the enterprise profits shown in Figure 13. Company A1 takes advantage of its slightly superior production costs and available resources. The profits for A1, A2, A3, B1 (secondary company), and C1 (tertiary company) are 3590, 2467, 3000, 5000, and 9681, respectively, given lower profit requirement levels of 0, 0, 0, 2000, and 4000. The λ multipliers take nonzero values for λ41 ) 0.803, λ51 ) 0.197, λ32 ) 0.533, λ42 ) 0.467, λ43 ) 1, λ44 ) 1, λ45 ) 0.080, and λ55 ) 0.920. The CPU requirement is 19 s, and the error at the solution point is 0.20%. It may seem unusual to have a lower computational time for this example which has a larger problem size than the previous one, but the CPU requirements have been found to be sensitive to the choice of the lower requirement point. 5.4. A Six-Enterprise Example. The six-enterprise example has the same properties as the five-enterprise example and an extra final producer, B2. The connectivity network is given in Figure 14. The enterprise profits are shown in Figure 15. The profits for A1, A2, A3, B1, B2, and C are 3000, 3085, 3031, 5000, 5000, and 6000, respectively, given lower profit requirement levels of 0, 0, 0, 1000, 1000, and 2000. The CPU requirement is 646 s, and the error at the solution point is 0.023%. The computational errors are generally small, especially considering that these are large-scale supply chain applications. The CPU time of the first example is higher because of the more complex nature of the

production problem. For the other examples, the CPU requirements vary because of the time to find an integer solution within the marginal gap (5%). 6. Concluding Remarks This paper has considered an approach to sustainable profit sharing in a n-enterprise supply chain. A separable programming approach, which utilizes the game theoretical, fair bargaining concepts developed by Nash and Harsanyi, has been presented. Semicontinuous transfer prices have been used in order to distribute profits between supply chain partners. The proposed model is a MINLP problem aiming to determine production resource utilization, production levels, inventory levels, flows, and transfer prices of the products in the supply chain network so as to maximize the profit levels of the separate enterprises fairly. The computational results show that the proposed method produces equitably distributed profits with low errors on the solutions. We believe there is a strong future for optimized multienterprise supply chain systems which utilize top level enterprise performance measures as objectives, through which the production process activities are planned directly. In this paper, we have described such a system. In the examples, the errors range from 0.013% to 0.60%. Our work assumes that all relevant data can be shared by all supply chain players, thus allowing the supply chain network to have a single optimization model. However, it is still quite common in practice that players are reluctant to make all information available to their partners. The development of suitable optimiza-

Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1659

tion systems characterized by private and public information constitutes our current research. Finally, it would be expected that real world size applications may be somewhat larger but still within the scope of the method. Although the approach is based on deterministic demand data, the key results to be retained for operational use are the transfer prices. Hence, representative demands can be used to determine these, and shorterterm operational scheduling tools can be used to manage the supply chain in practical situations of dynamic demands. Acknowledgment The authors of this paper express their gratitude to Professor C. C. Pantelides for his suggestions and helpful discussions. Appendix: A Separable Programming Approach Consider a continuous strictly convex function in one variable, f(x), to be minimized. This function can be approximated over an interval as a piecewise linear function; see Figure 4. Using n grid points, the function f can be linearly approximated as16

The overall problem then becomes to maximize the following function: m

m

∑ λk f(xk) k)1

∑ ∑ λkjgij(xkj) e pi j)1 k)1 ∑ λkj ) 1 λkj g 0

m

max Φ ˆ )

m

∑

max fj(x) x j)1

(20)

m

subject to

m

gij(xj) e pi ∑ j)1

∀i

(21)

xj g 0

∀j

(22)

can be approximated over the n grid points (not necessarily equidistant). P have the properties that the fj functions are strictly concave (remember that maximization of a concave function has the same properties as minimization of a convex one) and gij are convex for i ) 1, ..., p. In the following function, ˆfj are the separable portions of the objective function

(29)

∀ k, j

(30)

n

λkj ln(πkj - πLkj) ∑ ∑ j)1 k)1

(31)

n

∑ ∑ λkjgij(xkj) e pi j)1 k)1

(19)

n The constraints ∑k)1 λk ) 1 and λk g 0 and the convexity requirement guarantee that two adjacent nodes take nonzero values. A nonlinear problem P with j separable functions to be optimized

∀j

The generalized Nash objective is to maximize the multiplied profits of the enterprises, i.e., Φ ) m (πj - πLj ) where m is the number of participating ∏j)1 companies, πj is the profit of company j, and πLj is the lower profit requirement point of company j. The objective function can be rewritten via logarithmic differentiation as ln Φ ) ∑jln(πj - πLj ). If the separable approach is utilized for the Nash problem, with logarithmic differentiation, the reformulated problem becomes

(18)

∀k

(28)

k)1

n

λk g 0

∀i

n

(17)

∑ λk ) 1 k)1

(27)

n

n

hf (x) )

n

∑ ∑ λkj f(xkj) j)1 k)1

max F ˆ )

∀i

(32)

n

∑ λkj ) 1

∀j

(33)

∀ k, j

(34)

k)1

λkj g 0

To return to the original notation introduced in section 3, the explicit change in the formulation is in constraint (10). A new objective function (36) is introduced. Also, the constraints relating to the λ values are introduced as constraints (37) and (38). It can be seen that the L ) convexity properties hold, because fm ) ln(πkm - πkm is maximized and is strictly concave (equivalent to minimization of a convex function) and πkm is linear and therefore convex. n

∑ λkmπkm ) k)1 ∑i ∑t (ηimSimt + m′∈T ∑h k∈TP ∑ Pˆ imm′k XFimm′kt) - Cm im

i

∀ m (35)

n

n

ˆfj )

∑ k)1

∀j

λkj f(xkj)

(23)

max Φ ˆ )

λkm ln(πkm - πLm) ∑ ∑ m k)1

(36)

n

m

n

∑ ∑ λkjgij(xkj) e pi j)1 k)1

∀i

(24)

λkm g 0

n

∑ λkj ) 1

∀m

(37)

∀ k, m

(38)

∀j

(25)

Literature Cited

∀ k, j

(26)

(1) Cox, A. Power, value and supply chain management. Supply Chain Manage. 1999, 4, 167-175.

k)1

λkj g 0

∑ λkm ) 1

k)1

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(2) van Hoek, R. I. Measuring the unmeasurablesmeasuring and improving performance in the supply chain. Supply Chain Manage. 1998, 3, 187-192. (3) Cohen, M. A.; Lee, H. L. Resource deployment analysis of global manufacturing and distribution networks. J. Manuf. Oper. Manage. 1989, 2, 81-104. (4) Arntzen, B. C.; Brown, C. G.; Harrison, T. P.; Trafton, L. L. Global supply chain management at Digital Equipment Corporation. Interfaces 1995, 25, 69-93. (5) Pfeiffer, T. Transfer pricing and decentralized dynamic lotsizing in multistage, multi-product production processes. Eur. J. Oper. Res. 1999, 116, 319-330. (6) Alles, M.; Datar, S. Strategic transfer prices. Manage. Sci. 1998, 44, 451-461. (7) Jose, R. A.; Ungar, L. H. Pricing interprocess streams using slack auctions. AIChE J. 2000, 46, 575-587. (8) Ballou, R. H.; Gilbert, S. M.; Mukherjee, A. New managerial challenges from supply chain opportunities. Ind. Mark. Manage. 2000, 29, 7-18. (9) Nash, J. The bargaining problem. Econometrica 1950, 18, 155-162. (10) Conley, J. P.; Wilkie, S. An extension of the Nash bargaining solution to nonconvex problems. Games Econ. Behav. 1996, 13, 26-38.

(11) Harsanyi, J. C. Rational Behavior and Bargaining Equilibrium in Games and Social Situations; Cambridge University Press: Cambridge, U.K., 1977. (12) Young, H. P. Equity; Princeton University Press: Princeton, NJ, 1994. (13) Gjerdrum, J.; Papageorgiou, L. G.; Shah, N. Fair transfer price and inventory holding policies in two-enterprise supply chains; Technical Report, Process Systems Engineering B99.18; Imperial College: London, U.K., 1999. (14) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R. GAMSs Language Guide; GAMS Development Corp.: Washington, DC, 1997. (15) CPLEX Optimization Inc. Using the CPLEX callable library; CPLEX: Incline Village, NV, 1998. (16) Bazaraa, M. S.; Shetty, C. M. Nonlinear Programming; John Wiley and Sons: New York, 1979.

Received for review July 18, 2000 Accepted January 7, 2001 IE000668M